Let X be a generic smooth irreducible complex projective curve of genus g withg≥4. In this paper, we generalize the existence theorem of special divisors to high dimensionalindecomposable vector bundles. We give a necessary and sufficient condition on the existence ofn-dimensional indecomposable vector bundles E on X with det(E)=d, dim H^0(X,E)≥h. Wealso determine under what condition the set of all such vector bundles will be finite and how manyelements it contains.
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